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The First Attempt at Reforming Mathematics

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Descartes’s Mathematical Thought

Part of the book series: Boston Studies in the Philosophy of Science ((BSPS,volume 237))

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Abstract

The inspiration which led Descartes to his reform of traditional mathematical thought took place soon after his encounter in the autumn of 1618 with Isaac Beeckman, a Dutch scholar his senior by eight years. The significance of this meeting for him is shown clearly in his letter of April 23, 1619 to Beeckman: “You are truly the only one who awoke [me] from sloth, recalled erudition which had almost passed away from memory, and bettered my mind which was drifting away from serious occupations.”1

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References

  1. “Descartes, à Bréda, à Isaac Beeckman, à Middlebourg. 23 avril 1619,” AT, X, pp. 162163; Journal tenu par Isaac Beeckman, éd. C. De Waard, t. IV (Supplément), (La Haye, 1953), p. 62: “Tu enim reverâ solus es qui desidiosum excitasti, jam è memoriâ penè elapsam eruditionem revocasti et à serijs occupationibus aberrans ingenium ad meliora reduxisti.”

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  2. Journal, t. IV, pp. 17–19.

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  3. Consult the “Table des ouvrages cites” and “Index général” attached to the end of Volume IV of Beeckman’s Journal.

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  4. For example, Beeckman writes “1a aequatur 41749Q, + 276128” for today’s æ2 = 4174x + 276128, in Journal,t. I (La Haye, 1939), p. 6.

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  5. See Dirk J. Struik, The Land of Stevin and Huygens (Dordrecht/Boston/London, 1981), p. 77.

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  6. See Walter J. Ong, Ramus, Method, and the Decay of Dialogue: From the Art of Discourses to the Art of Reason (Cambridge, Mass., 1958), p. 305; P. Dibon, “L’influence de Ramus aux universités néerlandaises du 17’ siècle,” Proceedings of the XIth International Congress of Philosophy, Vol. XIII (Amsterdam, 1953), pp. 307–311.

    Google Scholar 

  7. As to the main characteristics of Beeckman’s natural philosophy, see R. Hooykaas, “Science and Religion in the Seventeenth Century: Isaac Beeckman (1588–1637),” Free University Quarterly, 1 (1951), pp. 169–183; Idem, “Beeckman, Isaac,” DSB, I, pp. 566–568; John A. Schuster, “Descartes and the Scientific Revolution, 1618–1634: An Interpretation” (Princeton University Ph. D. Dissertation, 1977), Chapter 1, B. “Beeckman and Descartes,” pp. 53–71.

    Google Scholar 

  8. “Pierre Gassendi, à Bruxelles, à Nicolas-Claude Fabri de Peresc, à Aix. 21 juillet 1629,” CM, II, p. 246. For Beeckman’s role in Gassendi’s intellectual career, see Howard Jones, Pierre Gassendi 1592–1655: An Intellectual Biography (Nieuwkoop, 1981), p. 27.

    Google Scholar 

  9. AT, X, pp.46–47; Journal, t. I, p. 237: “{Angulum nullum esse malè probavit Des Cartes.} Nitebatur heri, qui erat 10 Nov., Breda Gallus Picto probare nullum esse angulum reverâ, hoc argumento: Angulus est duarum linearum concursus in uno puncto, ut ab et cb in puncto b. At si seces angulum abc per lineam de, divides punctum b in duas partes, ita ut ejus dimidium ab adjungatur, alterum dimidium bc. Quod est contra puncti definitionem, cui pars nulla. At ille punctum sumpsit pro reali magnitudine, cùm punctus nihili aliud sit quàm extremitas linea ab et cb. Nec totum complet punctus, ita ut mille puncti possent esse eodem loco. Linea igitur de transit per punctum quidem b, sed id non secat, verùm totum complet, cùm linea non sit lata. Quare punctum aliquod in lineâ de eodem in loco est quo punctum b. Tale etiam punctum est fg. Non igitur linea f g, de, secantes angulum, minuunt lineas ab et cb, ut fit cùm serrâ quid secamus, sed solummodo separant unam ab aliâ.” A margin headline is shown with the parentheses.

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  10. Clavius, Euclidis Elementa, Opera mathernatica,t. I, p. 13: “Pvnctvm est, cuius pars nulla est.”

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  11. As a contemporary book related to the above-mentioned problem, the editor C. de Waard refers to Henri de Monantheuil, De Puncto primo geometriae principio Liber (Lugd. Bat., 1600). Journal, t. I, p. 237, n. 4).

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  12. AT, X, p. 54; Journal, t. I, p. 255: “Renatus Descartes mihi proposuit problema: Dare quadratum rquale radici alterius quadrati.”

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  13. AT, X, p. 55; Journal, t. I, p. 255: “Ut se habet 9 ad 1 sic ab ad e; sed cd est medium proportionale inter ab et e, ergo est latus secundi quadrati.”

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  14. A Koyré, Études galiléennes, (Paris, 1939; actually published in 1940), pp. 99–128 (II-25–54). Koyré assigned the erroneous mathematical formulation of the law of free fall to Descartes’s one-sidedness as a pure mathematician. A more historically sound analysis may be seen in J. A. Schuster, Op. cit. (n. 7), pp. 72–93. Further, on the development of Descartes’s natural philosophy from 1618 on, see Vincent Jullien et André Charrak, Ce qui dit Descartes touchant la chute des graves: 1618 à 1646, étude d’un indicateur de la philosophie naturelle cartésienne (Villeneuve d’Ascq, 2002).

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  15. AT, X, p. 52; Journal, t. I, p. 244: “{Physico-mathematici paucissimi.} Hic Picto cum multis Jesuitis alijsque studiosis virisque doctis versatus est. Dicit tamen se nunquam neminem reperisse, praeter me, qui hoc modo, quo ego gaudeo, studendi utatur accuratèque cum Mathematicâ Physicam jungat. Neque etiam ego, praeter ilium, nemini locutus sum hujusmodi studij.”

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  16. “Descartes, à Breda, à Isaac Beeckman, à Middlebourg. 26 mars 1619,” AT,X, p. 154; Journal, t. IV, p. 58: “Quatuor enim à tam brevi tempore insignes et planè novas demonstrationes adinveni, meorum circinorum adjumento.”

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  17. AT, X, pp. 154–156; Journal, t. IV, pp. 58–59: “Prima est celeberrima de dividendo angulo in æquales partes quotlibet. Tres aliæ pertinent ad æquationes cubicas, quarum primum genus est inter numerum absolutum, radices et cubos, alterum inter numerum absolutum, quadrata et cubos, tertium denique inter numerum absolutum, radices, quadrata et cubos. Pro quibus tres demonstrationes repperi, quarum unaquque ad varia membra est extendenda propter varietatem signorum + et —; quæ omnia nondum discussi, sed facilè, meo judicio, quod in unis repperi, ad alia applicabo. Atque hac arte quadruplo plures quaestiones et longè difficiliores solvi poterunt quàm communi Algebrâ; tredecim enim diversa genera æquationum cubicarum numero, qualia tantùm sunt tria æquationum communium, nempe inter 1a et o%C+oN, vel o2C—oN, vel denique oN—o2C. Aliud est quod jam quæro de radicibus simul ex pluribus varijs nominibus compositis extrahendis, quod, si reperero, ut spero, scientiam illam planè digeram in ordinem, si desidiam innatam possim vincere et fata liberam vitam indulgeant.”

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  18. Cf. Gustav Eneström’s remarks in AT, X, p. 155, nn. a. and d.

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  19. Cf. G. Eneström’s note a. in AT, X, p. 156.

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  20. Clavius, Algebra, Cap. 28, “De Extractione Radicvm ex Binomiis, et Apotomis. Vbiobiter de alijs lineis Irrationalibus, de quibus Euclides in lib. 10. disputat.” Opera mathematica, t. II, pp. 72–79.

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  21. AT, X, pp. 156–158; Journal, t. IV, pp. 50–60: “[Beeckman: {Ars generalis ad omnes quaestiones solvendas quæsita.}] Et certè, ut tibi nudè aperiam quid moliar, non Lullij Artem brevem, sed scientiam penitus novam tradere cupio, quâ generaliter solvi possint quæstiones omnes quæ in quolibet genere quantitatis, tàm continuæ quàm discretæ, possunt proponi. Sed unaquæque juxta suam naturam: ut enim in Arithmeticâ quædam quæstiones numeris rationalibus absolvuntur, aliæ tantùm numeris surdis, aliæ denique imaginari quidem possunt, sed non solvi, ita me demonstraturum spero, in quantitate continuâ, qudam problemata absolvi posse cum solis lineis rectis vel circularibus; alia solvi non posse nisi cum alijs lineis curvis, sed quæ ex unico motu oriuntur, ideòque per novos circinos duci possunt, quos non minus certos existimo et geometricos quàm communis quo ducuntur circuli; alia denique solvi non posse nisi per lineas curvas ex diversis motibus sibi invicem non subordinatis generatas, quæ certè imaginari tantùm sunt: talis est linea quadratrix, satis vulgata. Et nihil imaginari posse existimo, quod saltem per tales lineas solvi non possit, sed spero fore ut demonstrem quales quæstiones solvi queant hoc vel illo modo et non altero, adeò ut penè nihil in Geometriâ supersit inveniendum. Infinitum quidem opus est, nec unius. Incredibile quàm ambitiosum, sed nescio quid luminis per obscurum hujus scienti chaos aspexi, cujus auxilio densissimas quasque tenebras discuti posse existimo.”

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  22. See Heath, A History of Greek Mathematics, Vol. 1, pp. 175–176.

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  23. Opere di Galileo Galilei, ed. Antonio Favaro (Firenze, 21932), t. II, p. 369; Translated, with an Introduction by Stillman Drake (Washington, D. C., 1978), p. 41.

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  24. Ibid.

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  25. Cf. S. Drake’s introduction in Op. cit., esp. pp. 26–29.

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  26. Benjamin Bramer, Beschreibung und Underricht, wie allerley Theylungen zu den Mathematischen Instrumenten zu verfertigen: Neben dem Gebrauch eines newen Proportional Instruments (Marburg, 1615), pp. 5 Sc 91. For Bramer’s sector, see No Schneider, “Der Proportionzirkel, ein universelles Analogrechnerinstrument der Vergangenheit,” Deutsches Museum Abhandlungen und Berichte, 38 (1970), Heft 2, pp. 1–96, esp. pp. 58–62. I am indebted to Prof. MIURA Nobuo in Kobe University for my understanding of how to construct and use Bramer’s sector.

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  27. ín Britain the sector has been distinguished from the proportional compass. “The European names for the sector are (German) Kreissektor; (Italian) compasso di proporzione; (French) compas de proportion. The proportional compass is known as (German) Proportionalzirkel; (Italian) compasso di reduzione; (French) compas de réduction.” Gerard L’E. Turner, Antique Scientific Instruments (Poole/Dorset, 1980), p. 58. See also Maurice Daumas, Scientific Instruments of the Seventeenth and Eighteenth Centuries,translated and edited by Mary Holbrook (London, 1972), pp. 22–25.

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  28. Clavius mentioned some mathematical instruments, for example, the ordinary compass and the quadrant in his Geometria practica, in Opera mathematica, t. II, Lib. I, Caput I, “Instrvmenti partivm constructio, atque vsus,” pp. 5–14. Cf. I. Schneider, Op. cit. (n. 26), p. 46.

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  29. “Descartes, à Amsterdam, à Isaac Beeckman, à Middlebourg,” AT, X, p.164; Journal, t. IV, p. 63: “Repperi nudius tertius eruditum virum in diversorio Dordracensi, cum quo de Lullij Arte parvâ sum loquutus.”

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  30. “Isaac Beeckman, à Middlebourg, à Descartes, à Copenhague. 6 mai 1619,” AT, X, p. 168; Journal, t. IV, p. 65. We are aided in understanding Beeckman’s description of Lull’s Ars brevis by Frances A. Yates, “The Art of Ramon Lull: An Approach to It through Lull’s Theory of Elements,” Journal of the Warburg and Courtauld Institutes, 17 (1954), pp. 115–173, esp. pp. 116–117; Reprinted in Eadern, Lull and Bruno: Collected Essays, Vol. I (London, 1982), pp. 9–77, esp. pp. 10–11. Cf. Heinrich Cornelius Agrippa von Nettesheim, In artem brevem Raymundi Lullij commentaria, in Operum pars posterior (Lyon, 1600; Hildesheim/New York, 1970).

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  31. AT, X, p. 65; Journal, t. I, p. 295: “{Ars Lullij cum Logica collata.} Particulares scientiae igitur sunt vice artis Lullianae, ars verò Lullij non potest planè esse vice Logicae.” The editor of Beeckman’s Journal remarks that Beeckman was taught the method of Ramus while studying at Leiden and that his teacher Rudolph Snel published the Commentaria in Dialecticam Petri Rami (Herborn, 1587).

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  32. AT, VI, p. 17

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  33. Agrippa, Op. cit. (n. 30). For Agrippa’s thought in general, see Richard H. Popkin, Introduction to Agrippa, Opera, I (Hildesheim/New York, 1970); Charles G. Nauert, Jr., Agrippa and the Crisis of Renaissance Thought, Illinois Studies in the Social Sciences (Urbana, 1965).

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  34. Kurt Vogel, “Stifel, Michael,” DSB, XIII, p. 58.

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  35. See Paolo Rossi, Clavis universalis: Arti della memoria e logica combi-natoria da Lullo a Leibniz (Bologna, 21983; first published in 1960), Cap. II, “Enciclopedismo e combinatoria nel Cinquecento,” pp. 63–102, Cap. V, “La memoria artificiale e il metodo della nuova scienza: Ramo, Bacone, Cartesio,” 1, “Pierre de la Ramée: la ((memoria» come sezione della logica,” pp. 155–162; Frances A. Yates, The Art of Memory (Chicago, 1966), Chap. VIII, “Lullism as an Art of Memory,” pp. 173–198, Chap. X, “Ramism as an Art of Memory,” pp. 231–242; Ong, Op. cit. (n. 6).

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  36. Paolo Rossi, “The Legacy of Ramon Lull in Sixteenth-Century Thought,” Medieval and Renaissance Studies, 5 (1961), pp. 184–185.

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  37. In his Clavis universalis, Rossi wrote a suggestive comment on the Descartes of this period: “It is certain that the problem of the young Descartes—a man who has not `caught his foundations of physics to be sided with’—may particularly appear to be close to the one in the Lullian syntax and encyclo-pedia of the late sixteenth century: Behind the manifold of sciences there hides a profound unity, a law of connection, a common logic.” Op. cit. (n. 35), p. 177 (my translation).

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  38. Op. cit. (n. 1), in AT, X, p. 163; Journal, t. IV, p. 62: “Quod ad cætera quæ in superioribus me invenisse gloriabar, verè inveni cum novis circinis, nec decipior. Sed membratim non ad te scribam, quia integrum opus hac de re meditabor aliquando, meo judicio novum, nec contemnendum.”

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  39. Charles Adam, “Avertissement” to “Opuscules de 1619–1621, Ms. de Leibniz,” AT, X, p. 207.

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  40. Foucher de Careil, vivres inédites de Descartes, t. I (Paris, 1859), pp. 1–57. 41 “Inventaire succinct des Écrits etc.,” AT, X, pp. 7–8.

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  41. Cogitationes privatae, in AT, X, p. 214: “Polybij Cosmopolitani Thesavrvs Mathematices, in quo traduntur vera media ad omnes hujus scientiae difficultates resolvendas, demonstraturque circa illas ab humano ingenio nihil vitra posse praestari: ad quorumdam, qui nova miracula in scientijs omnibus exhibere pollicentur vel cunctationem provocandam & temeritatem explodendam; turn ad multorum cruciabiles labores sublevandos, qui, in quibusdam hujus scientiae nodis Gordijs noctes diesque irretiti, oleum ingenij inutiliter absumunt: totius orbis eruditis & specialiter celeberrimis in G. (Germaniâ) F. R. C. denuo oblatus.” The translation is taken from The Philosophical Writings of Descartes, translated by John Cottingham, Robert Stoothoff and Dugald Murdoch, Vol. 1 (Cambridge, 1985), p. 2.

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  42. H. Gouhier, Les Premières Pensées de Descartes: Contribution à histoire de l’AntiRenaissance (Paris, 1958), p. 109.

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  43. Rheticus-Pitiscus, Thesaurus mathematicus sive Canon sinuum ad radium 1.00000.00000.00000 (Francifurti, 1613). For this work, see R. C. A[rchibold], “Notes,” Mathematical Tables and Other Aids to Computations, 3 (1948–1949), p. 558. See also Edward Rosen, “Rheticus, George Joachim,” DSB, XI, pp. 395–398; H. L. L. Busard, “Pitiscus, Bartholomeo,” DSB, XI, pp. 3–4. As to Pitiscus’s position in the history of trigonometry, I have learn much from Nobuo Miura, “The Application of Trigonometry in Pitiscus: A Preliminary Essay,” Historia Scientiarum, No. 30 (1986), pp. 63–78.

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  44. On the relation between Descartes and the Rosicrusian movement, see Frances A. Yates, The Rosicrusian Enlightenment (London, 1972), Chap. VIII, “The Rosicrusian Scare in France,” pp. 103–117. Yates’s description is almost totally dependent upon A. Baillet, La Vie de Monsieur Des-cartes (Paris, 1691) and must be read critically. For criticism of Yates’s general historiography emphasizing the continuity between the Renaissance and the Scientific Revolution, see P. Rossi, “Hermeticism, Rationality and the Scientific Revolution,” in M. L. Righini Bonelli and William Shea, eds., Reason, Experiment and Mysticism in the Scientific Revolution (New York, 1975), pp. 247–273. Gouhier, on the other hand, places stress on the discontinuity between the two periods in his Les Premières Pensées de Descartes: Contribution à l’histoire de l’Anti-Renaissance (n. 43). He uses the term `Anti-Renaissance’ as the French synonym for `Counter-Renaissance’. The latter nomination was first used by Hiram Haydn in his Counter Renaissance (New York, 1950) concerning the history of ideas of the Elizabethan period. Haydn defined the historical concept `Counter-Renaissance’ as “a protest against the basic principles of the classical renaissance, as well as against those of medieval Scholasticism.” Op. cit., Introduction, p. xi.

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  45. H. Gouhier, Op. cit.(n. 43), p. 110.

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  46. AT, X, p. 214; Transtation, Op. cit. (n. 42), p. 2.

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  47. AT, X, p. 214: “Vt comcedi, moniti ne in fronte appareat pudor, personam induunt: sic ego, hoc mundi theatrum conscensurus, in quo hactenus spectator exstiti, larvatus prodeo.” 49AT, X, p. 215.; Translation, Op. cit. (n. 42), p. 3.

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  48. AT, X, pp. 234–235: “Inveni aquationes inter talia: 1Ce & 72C+ 14, & simile hoc. Reduco ad 126 + 2 aqu. CC, & quaro 1CC, quem postea multiplicabo per 7 [primi circini]. Deinde alium circinum habere oportet, quorum dua partes sunt tales. Prima habet lineam be firmiter annexam ad angulos rectos linea a f, lineam autem de ad angulos quidem rectos, sed mobilem per lineam f b. Linea fb habet praterea in puncto d stylum fixum, quo aliam lineam describit; in puncto f etiam vnum, sed mobilem, quo aliam lineam describit hoc pacto. Secunda pars dcegh, constans lineis firme invicem annexis, fluat supa lineam ap, vbi affixa est prima pars in puncto a immobili: punctum c impellit lineam dc,& ita efficiet vt tota secunda pars descendat, linea autem cd trahit lineam de per spatium fb juxta varietatem intersectionum, & tum stylus d lineam primi circini describet. Linea autem gh intersecabit etiam lineam de, aliamque lineam curvam stylo c mobili describet, quw vltima linea secabit ap, in quo ae est cubus inveniendus, si ab primes partis sit vnitas, ce verò secundæ numerus absolutus, qui in exemplo est binarius.” Here we draw attention to cossic symbols. Compared with Leibniz’s autograph copy of the manuscript De solidorum elementis (see § 3), Leibniz is supposed to have used his own symbols 21. (the astronomincal sign for Jupiter) and 9 (C with a little bar) for %e and CC, respectively. See Descartes, Exercices pour les éléments des solides, éd. P. Costabel (Paris, 1987), Costabel’s Introduction, p. x. But it is certain that Descartes’s were the ordinary cossic ones 9C and CC as can be imagined by his letters to Beeckman. Thus, we do not rewrite the cossic symbols into Leibniz’s.

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  49. AT, X, p. 239. On Descartes’s compasses in general, see Michel Serfati, “Les compas cartésiens,” Archives de philosophie, 56 (1993), pp. 197–230. See also Idem, “Le Développement de la pensée mathématique du jeune Descartes (l’éveil d’un mathématicien),” in De la Méthode: Recherches en histoire et philosophie des mathématiques, dirigé par M. Serfati (Besançon/Paris, 2002), pp. 39–104, which has provided a useful discussion on Descartes’s compasses against the background knowledge on the mathematical career of young Descartes.

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  50. AT, VI, pp. 391–392; Olscamp, pp. 191–192.

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  51. We refer to G. Eneström’s note b. in AT, X, p. 235.

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  52. AT, X, p. 236: “Fit præterea æquatio inter talia, Ce, a’, 2e, dummodo quot sint a’ tot 2e, & hoc modo: 1CC æqu. 6$-62C+56. Deinde ex N tollo vnitatem, ex residuo cubum formo, cujus radici vnitatem addo, & quod cubice producitur ex illâ radice est z Ce; quod si multiplicetur per 2, producet cubum quaesiturn.”

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  53. AT, X, p. 236, note a.

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  54. Op. cit., p. 237: “Addo vnitatem numero absoluto; deinde ex radice producti vnitatem demo, & producitur ex radice cubus quæsitus.”

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  55. AT, X, pp. 238–239: “Alios circinus ad ßquationes 1 Ce l4 O8 ON. Si inveniendus sit cubus æqualis ON dg & quadrato vni incognito, talis circinus fabricetur: dce Huit supra ap, fluendo pellit bc in puncto c adigitque vt descendat simulque af,cui affixa est bc ad angulos rectos, describitque intersectione a f & cd lineam circini mesolabi.”

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  56. ee Leibniz’s remarks c, d, & f on p. 239.

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  57. P. 239, note a. soAT, X, p. 240.

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  58. AT, X, p. 244: “Regula generalis ad æquationes quatuor terminorum completas.”

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  59. Pappus, Collectio, ed. F. Hultsch, I (Berlin, 1876), pp. 54–56; La Collection mathématique, tr. Ver Eecke, t. I (Paris, 1933), p. 39.

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  60. Collectiones, ed. F. Commandino (Pesaro, 1588), f. 4v.

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  61. Archimedes, Opera omnia cvm Commentariis Evtocii, ed. I. L. Heiberg (corrigenda adiecit E. S. Stamatis), Vol. III (Stvtgardiae, 1972), pp. 88–97; Archimède, tr. Charles Mugler, t. IV: Commentaires d’Eutocius et fragments (Paris, 1972), pp. 64–69; Les OEuvres complètes d’Archimède, tr. Paul Ver Eecke, t. II (Paris, 1960), pp. 609–615.

    Google Scholar 

  62. Marshall Clagett, Archimedes in the Middle Ages, Vol. II: The Translations from the Greek by William of Moerbeke, Part II: Texts (Philadelphia, 1976), pp. 246–248. 66Commandino, Op. cit.(n. 63), f. 5r.

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  63. Vitruvius, De Architectura, Lib. 9, Prooem. 14: “Itaque Archytas cylindrorum descriptionibus, Eratosthenes organica mesolabi ratione idem explicaverunt.” I. e. “Archytas solved the problem [the duplication of the cube] by a diagram with cylinders; Eratosthenes by means of an instrument, the mesolabium.” See Vitruvius, On Architecture, Vol. II, tr. Frank Granger (London 1934), pp. 206–207.

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  64. Commandino (n. 63), Lib. III. Problema I. Propositio V. f. 5r: “Duabus datis rectis lineis, duas medias proportiones in continua analogia inuenire.”

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  65. Heath, A History of Greek Mathematics, Vol. 1, pp. 252–253.

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  66. Commandino, f. 5r-5v. Cf. Heath, Vol. 1, pp. 258–259.

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  67. AT, VI, pp. 442–443; Olscamp, p. 228. Descartes’s mathematical demonstration which follows is: “For if we wish to find two mean proportionals between YA and YE, we have only

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  68. Clavius, Geometria practica (n. 28), pp. 160–163. Almost all the mathematicians referred to here are examined in Heath, A History of Greek Mathematics, Vol. 1, pp. 244–270. As for what was known during the Renaissance on the problem of mean proportionals, see Marshall Clagett, Archimedes in the Middle Ages, Vol. III: The Fate of the Medieval Archimedes, 1300 to 1565, Part III: The Medieval Archimedes in the Renaissance, 1450–1565 (Philadelphia, 1978), pp. 1163–1179.

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  69. Principally only two ancient authors, Vitruvius and Pappus, are referred to in Henricus Stephanus, Thesaurus Graecae Linguae, Vol. VI (Graz, 1854), col. 616 and in Henry George Lidell and Robert Scott, A Greek-English Lexicon (Oxford, 1968), p. 1106. There could have been other cases, of course.

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  70. J. J. Scaliger, Mesolabium (Leiden, 1594); Viète, Variorum de rebus Mathematicis Responsorum Liber VIII (Tours, 1593); Idem, Pseudo-Mesolabum (Paris, 1595), in Opera math-ematica, ed. Frans van Schooten (Leiden, 1646), pp. 347–435 & 258–285. In his work, Viète mentioned Eutocius’s commentary on Archimedes and Vitruvius.(Opera mathematica, pp. 348–350.)

    Google Scholar 

  71. AT, X, pp. 240–241: “Circinus ad angulum in quotlibet partes dividendum. Sit talis circinus: ab, ac, ad, ae sunt æquales laminæ divisæ pariter in punctis f, i, k, litem f g æquales a f, &c. Vnde sit vt anguli, bac, cad & dae sint semper æquales, nec vnus possit augeri vel minui, quin alij etiam mutentur. Sit igitur angulus bax dividendus: applico lineam ae supra axquâ ibi manente immobili, elevo lineam ba in partem b, qu secum trahit ac & ad, lineaque describetur à puncto g talis ryse. Deinde sumatur na æqualis af, ex puncto n ducatur pars circuli 06o, ita vt nO sit etiam æqualis f g: dico lineam ab dividere angulum in tres partes æquales. Ita potest dividi angulus in plures, si circinus constet pluribus laminis.”

    Google Scholar 

  72. To use anachronistic conceptions and notation, the equation of the curve thy in polar coordinates is p = 2acos(sp/2), where p = a6, p = L6a-y,a = af, since p = ag = ah = cry cos L ha-y (L ha-y is a right triangle.) Cf. AT, X, p. 241, note a by Eneström.

    Google Scholar 

  73. AT, X, pp. 232–233: “Describi potest sectio conica tali circino: sit AD perpendicularis, superficies obliqua AB. Sit pes circini immobilis, volvatur BC supra planum obliquum, ita tarnen vt CB possit brevior fieri, si imaginetur per C ascendere.”

    Google Scholar 

  74. AT, X, p. 233: “Sectio cyclindri, eodem pacto, circino duci potest ita: sit ACDE circinus, cujus pes immobilis est; linea DE descendet vel ascendet libere per punctum D prout à plano distabit.”

    Google Scholar 

  75. AT, X, p. 241: “Vidi commodum instrumentum ad picturas omnes transferendas: constat in pede cum circino bicipiti.”

    Google Scholar 

  76. See G. L’E. Turner, Op. cit. (n. 27), p. 57.

    Google Scholar 

  77. AT, X, p. 242: “Aliud quoque ad omnia horologia depingenda, quod per me possum invenire. Tertium ad angulos solidos metiendos. Quartum argenteum ad plana & picturas metiendas. Pulcherrimum aliud ad picturas transferendas. Aliud affixum oratoris tibia ad momenta metienda. Aliud ad tormenta bellica noctu dirigenda.—Petri Rothen Arithmetica philosophica.—Benjamin Bramerus.”

    Google Scholar 

  78. Peter Roth, Arithmetica philosophica, oder […] Kunstliche Rechnung der Coß oder Algebrae (Nürnberg, 1608). Cf. AT, X, p. 242, note a. Charles Adam has recorded the year of publication as 1607 instead of 1608 by mistake.

    Google Scholar 

  79. B. Bramer, Op. cit.(n. 26), p. 39. As for the other works of Bramer, see Paul Kirchvogel, “Bramer, Benjamin,” DSB, II, p. 419.

    Google Scholar 

  80. S. Drake, Op. cit. (n. 23), Introduction, p. 22.

    Google Scholar 

  81. F. A. Yates, The Art of Memory (n. 35), p. 300. Cf. Biographie universelle (Michaud) ancienne et moderne, nouvelle éd. t. 38 (Paris, 1843), “Schenkel (Lambert/Thomas), 1547—c. 1603,” pp. 283–284.

    Google Scholar 

  82. AT, X, p. 230. The translation has been taken from Yates, Ibid., p. 373.

    Google Scholar 

  83. Rossi, Clavis universalis (n. 35), pp. 174–175.

    Google Scholar 

  84. AT, X, p. 223: “Petijt à me Isaacus Middelburgensis an funis ACB affixus clavis a, b, sectionis conic partem describat.”

    Google Scholar 

  85. For the history of the catenary problem, see H. J. M. Bos, “Newton, Leibniz and the Leibnizian Tradition,” Chap. 2 of I. Grattan-Guiness, ed., From the Calculus to Set Theory, 1630–1910 (London, 1980), pp. 80–82; See also Hermann H. Goldstine, A History of the Calculus of Variations from the 17th through the 19th Century (New York/Heidelberg/Berlin, 1980), Index “Heavy chain problem.”

    Google Scholar 

  86. AT, X, p. 223: “Quod non licet per otium nunc disquirere.”

    Google Scholar 

  87. AT, X, p. 241: “Si subtrahatur numeri triangularis quadratus ex quadrato sequentis triangularis, restat cubus. Vt 10, 15: tolle 100 ex 225, restat 125. Ex progressione 1 2 II 4 8 II 16 I 32 II habentur numeri perfecti 6, 28, 496.”

    Google Scholar 

  88. Heath, A History of Greek Mathematics, Vol. 1, pp. 76–84.

    Google Scholar 

  89. /bid., pp. 74–76.

    Google Scholar 

  90. Cf. Heath, The Thirteen Books of Euclid’s Elements,Vol. 2, (New York, 21956), pp. 421426; Clavius, Euclidis Elementa,pp. 378–379.

    Google Scholar 

  91. AT, X, pp. 246–247: “In tetraedro rectangulo, basis potentia ßqualis est potentijs trium facierum simul. V. g., sint basis tria latera, f, 20, 20; tria verò latera supra basin, 4, 2, 2: area basis erit 6; trium facierum, 2, 4, 4; quorum quadrata sunt, 36, <&> 4, 16, 16, quæ tria æquipollent priori. Item, sint latera basis 20, 5; & supra basin, 2, 3, 4: area basis erit 61; facierum verò, 3, 4, 6, quorum quadrata sunt 61, & 9, 16, 36, æqualia priori. Hinc plurimæ quæstiones ignotæ solvi possunt circa tetrædra rectangula & non rectangula per relationem ad rectangula. Heec demonstratio ex Pythagoricâ procedit, & ad quantitatem quoque quatuor dimensionum potest ampliari; in quâ quadratum solidi angulo recto oppositi æquale est quadratis ex 4 alijs solidis simul. Sit ad hoc paradigma processionum in numeris 1, 2, 3, 4; in figuris, 2e, 3, Ce; in angulis rectis duarum linearum, trium, quatuor.”

    Google Scholar 

  92. For a discussion of this problem, see Pierre Costabel, “L’Initiation mathématique de Descartes,” Archives de philosophie, 46 (1983), pp. 643–644. It is known that in the middle of the 19th century Ludwig Schläfli successfully extended the Pythagorean theorem to the multidimensional case. See his “Theorie der vielfachen Kontinuität,” Gesammelte mathematische Abhandlungen, Bd. 1 (Basel, 1950), p. 193.

    Google Scholar 

  93. AT, X, p. 247: “Data basi pyramidis rectangulæ, facilè inveniuntur latera super basin. Sint, v. g., latera basis,13, 20, &5. Pro primo latere supra basin ponatur 12C; pro altero, V.13 — 1I.; & pro tertio, x/.20 — 18; quorum duorum potentia, quia æqualis potentiæ lateris, est qualis 33–2k, vel 13’ æq. 4. Ergo nota basi & angulo opposito, totam pyramidem possumus agnoscere, vt de triangulo Euclides demonstrat.”

    Google Scholar 

  94. AT, X, pp. 247–248.

    Google Scholar 

  95. Clavius, Algebra, p. 5.

    Google Scholar 

  96. AT, X, p. 248. In our notation (or precisely speaking, in the later Descartes’s!), the area of the three sides are

    Google Scholar 

  97. Foucher de Careil, CEuvres inédites de Descartes, t. 2 (Paris, 1860), pp. 214–234.

    Google Scholar 

  98. For C. Mallet’s and E. Prouhet’s critical reviews, see Pasquale Joseph Federico, Descartes on Polyhedra: A Study on the De Solidorum Elementis (New York/Heidelberg/Berlin, 1982), pp. 6–7.

    Google Scholar 

  99. E. de Jonquières, “Écrit posthume de Descartes. De Solidorum Elementis. Texte latin (original et revu) suivi d’une traduction française avec notes,” Mémoires de l’Académie des Sciences de l’Institut de France, 2e série, 45 (1890), pp. 325–379.

    Google Scholar 

  100. Federico, Op. cit. (n. 103).

    Google Scholar 

  101. René Descartes, Exercices pour les éléments des solides: Progymnasmata de solidorum elementis—Essai en complément d’Euclide, Edition critique avec introduction, traduction, notes et commentaires par Pierre Costabel (Paris, 1987). See my review of this book in Isis, 79 (1988), pp. 731–732.

    Google Scholar 

  102. This partition which we follow below is due to Federico. In his edition (n. 106), Costabel calls the table of formulae at the end of the treatise “la troisième partie,” but does not consider it independently of Part II on polyhedral numbers.

    Google Scholar 

  103. Heath, The Thirteen Books of Euclid’s Elements, Vol. 3 (New York, 21956), pp. 261 & 267–268. Cf. Clavius, Euclidis Elementa, p. 479.

    Google Scholar 

  104. AT, X, p. 265; Federico, Op. cit., pp. 43–44.

    Google Scholar 

  105. G. Pólya, Induction and Analogy in Mathematics, Volume 1 of Mathematics and Plausible Reasoning (Princeton, 1954), pp. 57 & 226.

    Google Scholar 

  106. Federico, p. 45.

    Google Scholar 

  107. AT, X, pp. 266–267; Federico, p. 50. As we have already stated in our note 50 above, throughout his manuscript copy of De solidorum elementis, Leibniz systematically used his own symbols 21. and 9 for `se and Ce, respectively. We safely suppose that Descartes’s original symbols were the usual cossic ones. So in our texts, we use 2e and Ce, not those of Leibniz. 113AT, X, p. 265, 1. 20-p. 266, 1. 4; Federico, p. 46.

    Google Scholar 

  108. See Federico, pp. 50–51.

    Google Scholar 

  109. P Costabel, “Le théorème de Descartes-Euler,” Studia Cartesiana, 1 (1979), p. 34; Reprinted in Idem, Démarches originales de Descartes savant (Paris, 1982), p. 24.

    Google Scholar 

  110. /bid., p. 31; Reprint, p. 21.

    Google Scholar 

  111. AT, X, p. 267, 1. 31—p. 268, 1. 1; Federico, p. 54.

    Google Scholar 

  112. I. Lakatos, Proofs and Refutations: The Logic of Mathematical Discovery, ed. by John Worrall and Elie Zahar (Cambridge, 1976), p. 6, footnote 1.

    Google Scholar 

  113. Costabel, Loc. cit.(n. 116).

    Google Scholar 

  114. Costabel, Exercices pour les éléments des solides (n. 106), p. 101.

    Google Scholar 

  115. Costabel, Ibid., pp. 93–96. Cf. Costabel, “Euler lecteur de Descartes,” Revue du XVIII siècle, 18 (1986), pp. 281–288.

    Google Scholar 

  116. Leonhard Euler, “Elementa doctrinae solidorum,” Novi commentarii academiae scientiarum Petropolitanae, 4 (1752/3), 1758: Opera omnia, Ser. prima, XXVI (Lausanne, 1953), p. 78: “In omnia solido hedris planis incluso aggregatum ex numero angulorum solidorum et ex numero hedrarum binario excedit numerum acierum.”

    Google Scholar 

  117. Costabel, Op. cit. (n. 115), p. 34; Démarches originales de Descartes savant, p. 24.

    Google Scholar 

  118. AT, X, p. 269; Federico, p. 92; Costabel, Exercices (n. 106), p. 33.

    Google Scholar 

  119. Federico, p. 93.

    Google Scholar 

  120. Costabel, Exercices, p. 36: “Le mot solida, qu’il faut bien traduire par solides pour éviter de faire trop de place dans la traduction à une interprétation, est sans aucun doute appelé par cette idée sous-jacente.” “Les solida sont essentiellement des configurations pondérées, qui s’opposent aux formes pures et immatérielles, mais qui sont aussi bien à deux dimensions qu’à trois.”

    Google Scholar 

  121. Federico, pp. 89 & 94.

    Google Scholar 

  122. AT, X, p. 275, U. 4–10; Federico, p. 108.

    Google Scholar 

  123. Cf. Federico, p. 99.

    Google Scholar 

  124. Leonard Eugene Dickson, History of the Theory of Numbers, Vol. 2: Diophantine Analysis (Washington, D. C., 1919; New York, 21966), P. 5.

    Google Scholar 

  125. There is no indication of the place of publication, but it was probably in Frankfurt am Main. We refer to a copy at the Stadtbibliothek Ulm.

    Google Scholar 

  126. Gaston Milhaud, Descartes savant (Paris, 1921), pp. 86–87.

    Google Scholar 

  127. Federico, p. 118.

    Google Scholar 

  128. Faulhaber’s books have been examined in No Schneider, Johannes Faulhaber (n. 158, below), Kurt Hawlitscheck, Johann Faulhaber 1580–1635: Eine Bliitezeit der mathematischen Wissenschaften in Ulm (Ulm, 1995), and A. G. Kästner, Geschichte der Mathematik, Bd. III (Göttingen, 1799; Hildesheim/New York, 1970), pp. 111–152. Cf. Paul A. Kirchvogel, “Faulhaber, Johann,” DSB, IV,pp. 549–553.

    Google Scholar 

  129. D. Lipstorp, Specimen Philosophiae Cartesianae (Lugduni Batavorum, 1653), pp. 78–80. Cf. AT, X, pp. 252–253; A. Baillet, La Vie de Monsieur Des-Cartes (Pais, 1691), Premiere Partie, pp. 67–70. Baillet’s description was totally dependent on Lipstorp’s.

    Google Scholar 

  130. Federico, p. 117.

    Google Scholar 

  131. Without examining the tract of Numerus figuratus, Prouhet and Milhaud mistakenly thought that this tract treated some polyhedral numbers. This error has been corrected by Federico, pp. 116–118.

    Google Scholar 

  132. Peter Roth, Arithmetica philosophica, ff. 126r-129v.

    Google Scholar 

  133. Op. cit. (n. 131), p. 15.

    Google Scholar 

  134. Pappus, ed. Hultsch (n. 62), pp. 350–358; Commandino, ff. 33v-34r; Ver Eecke, pp. 272277.

    Google Scholar 

  135. Johannes Kepler, Harmonice mundi (Linz, 1619), in Gesammelte Werke, Bd. VI (München, 1940), pp. 83–87; L’Harmonie du monde, tr. Jean Peyroux (Paris, 1979), pp. 7480.

    Google Scholar 

  136. AT, X, r. 270; Federico, p. 96; Costabel, Exercices, p. 4.

    Google Scholar 

  137. AT, X, p. 274; Federico, p. 106; Costabel, Exercices, pp. 5–6. 144AT, X, p. 276; Federico, p. 108; Costabel, pp. 7 & 35.

    Google Scholar 

  138. Costabel, Exercices, pp. 87–92.

    Google Scholar 

  139. G. Milhaud, Op. cit. (n. 132), pp. 84–88.

    Google Scholar 

  140. Federico, p. 32.

    Google Scholar 

  141. Federico, p. 39. Cf. Albert Girard, Invention nouvelle en l’algebre (Amsterdam, 1629); Reprinted edition by D. Bierens de Haan (Leiden, 1884), “De la mesure de la superface des triangles & polygones spheriques, nouvellement inventé,” ff. Gv-[114]v.

    Google Scholar 

  142. Federico, pp. 32 & 39.

    Google Scholar 

  143. This note was first published posthumously as a part of `Excerpta ex Mss. R. Des-Cartes,“ in Opuscula posthuma, physica et mathematica (Amsterdam, 1701), p. 4. Cf. AT, X, pp. 297–299. Notice that the title ”Numeri polygoni“ was an addition by the editor of the Opuscula. In the 1701 Opuscula version, the cossic signs 2e and 3’ were rewitten as x and xx, respectively.

    Google Scholar 

  144. “Descartes, à (Santpoort), à Mersenne, à Paris. (3 juin 1638),” AT, II, pp. 158–168; AM, II, pp. 278–285; CM, VII, pp. 257–267. “Descartes, à (Santpoort), à Mersenne, à Paris. 27 juillet 1638,” AT, II, pp. 25–257; AM, II, pp. 354–355; CM, VII, pp. 405–407. Cf. L. E. Dickson, Op. cit.(n. 130), pp. 6–7.

    Google Scholar 

  145. AT, X, p. 271; Federico, p. 95.

    Google Scholar 

  146. Costabel, Exercices, pp. 103–109. I have arrived at my conclusion independently of Costa-bel.

    Google Scholar 

  147. Baillet, La Vie de Monsieur Des-Cartes, t. 1, pp. 80–86.

    Google Scholar 

  148. Frédéric de Buzon, “Un exemplaire de la Sagesse de Pierre Charron offert à Descartes en 1619,” Bulletin cartésian XX: Archives de philosophie, 55 (1992), pp. 1–3: “Doctissiomo Amico grato et minori fratri Renato Cartesio d. d. ded. P. Johannes B. Molitor S. J. exeunte Anno 1619 JBM.” Cf. G. Rodis-Lewis, Descartes (Introduction, n. 1), pp. 71 & 330, n. 42. The theory to ascribe the location of the poèle to Neuburg began with A. Baillet, Vie de Monsieur Descartes, rédite et abregé (Paris, 1692): Collection “Grandeurs” (Vanves, 1946), p. 33.

    Google Scholar 

  149. Kepler, Gesammelte Werke, Bd. XVII: Briefe 1612–1620 (München, 1955), p. 416: “Nescio utrum Cartelius quidam, vir sané eruditus, et humanitate singularj, meas tibj reddiderit: Per equidem invitus amicos molestiâ illâ afficio, quam interdum ipsis afferunt ingrati et impudentes aeruscatores. Cartelius tarnen alius videbatur, quemque consilio juvares dignissimus.” A photograph copy of the original letter can be seen in K. Hawlitscheck, Johann Faulhaber (n. 134), p. 59.

    Google Scholar 

  150. Hawlischeck, Loc. cit. (n. 134). Cf. Schneider (n. 158, below), the Edior’s note to the letter in Gesammelte Werke, Bd. XVII, p. 516, and Kenneth Manders, “Descartes et Faulhaber,” Archives de philosophie, 58 (1995), Bulletin cartésien XXIII, pp. 1–12.

    Google Scholar 

  151. Ivo Schneider, Johannes Faulhaber 1580–1635: Rechenmeister in einer Welt des Umbruchs (Basel/Boston/Berlin, 1993).

    Google Scholar 

  152. F. A. Yates, The Rosicrusian Enlightenment (n. 45), pp. 114–115. Cf. Hawlischeck, Op. cit. (n. 134), pp. 50–54.

    Google Scholar 

  153. Schneider, Op. cit. (n. 158), pp. 123–129.

    Google Scholar 

  154. Hiram Haydn, The Counter Renaissance (n. 45). See also H. Gouhier, Les Premières Pensées de Descartes: Contribution à l’histoire de l’Anti-Renaissance (n. 43).

    Google Scholar 

  155. Discours de la méthode, in AT, VI, p. 18. 163AT, VI, p. 19.

    Google Scholar 

  156. 7bid., rp. 19–20.

    Google Scholar 

  157. AT, V, pp. 176–177: “[…] requiritur ad id ingenium mathematicum, quodque usu poliri debet. Ea autem haurienda ex Algebrà.” See Descartes’s Conversation with Burman, translated with Introduction and Commentary by John Cottingham (Oxford, 1976), pp. 47–48.

    Google Scholar 

  158. AT, VI, pp. 20–21. 167/bid., pp. 21–22.

    Google Scholar 

  159. Jules Vuillemin, Mathématiques et métaphysique chez Descartes (Paris, 1960), Chapitre IV: “La théorie des proportions,” pp. 115–117.

    Google Scholar 

  160. Jean Dhombres, “Une Mathématique baroque en Europe: réseaux, ambitions at acteurs,” in L’Europe mathématique: Histoires, mythes, identités, sous la direction de Catherine Goldstein, Jeremy Gray et Jim Ritter (Paris, 1996), pp. 144–181, at p. 159.

    Google Scholar 

  161. Alice Browne, “Descartes’s Dream,” Journal of the Warburg and Courtauld Institutes, 40 (1977), pp. 256–273. In this regard, Sebba fundamentally agrees with Browne. See Gregor Sebba, The Dream of Descartes, Assembled from Manuscripts and Edited by Richard A. Watson (Carbondale and Edwardsville, Ill., 1987), p. 7: “What Descartes discovered was his mission, not his philosophy.” Cf. John R. Cole, The Olympian Dreams and Youthful Rebellion of René Descartes (Urbana/Chicago, 1992).

    Google Scholar 

  162. AT, X, p. 216; Translation (n. 42), p. 4.

    Google Scholar 

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Sasaki, C. (2003). The First Attempt at Reforming Mathematics. In: Descartes’s Mathematical Thought. Boston Studies in the Philosophy of Science, vol 237. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1225-5_4

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